HalfP: RP With Exactly Half Acceptance

Significantly, the number of candidate witnesses is restricted to be a power of 2. (This is implicit if they are binary strings.)

Contained in RP, EP, and ModkP for every odd k. Contained in EQP by the Deutsch-Jozsa algorithm.

Defined in [BB92], where it was called C==P[half]. The name used here is from [BS00]. There it was shown that HalfP is contained in every similar class in which 1/2 is replaced by some other dyadic fraction.

For functions and , we say that tuple , where is a language and is a distribution of problem instances, if there exists a heuristic deterministic algorithm such that for all in the support of , runs in time bounded by and with failure probability bounded by [BT06].

HO: High-Order logic

High order logic is an extension of Second order, First order where we add quantification over higher order variables.

We define a relation of order and arity to be a subset of -tuple of relation of order and arity . When it is by extension a first order variable. The quantification of formula in HO is over a given order (which is a straightforward extension of SO where we add quantification over constant (first-order variable) and relation (second-order variables). The atomic predicates now can be general application of relation of order and arity to relations of order and arity and test of equality between two relations of the same order and arity.

is the set of formulae with quantification up to order O. (resp. ) is defined as the set of formula in beginning by an existantial (resp universal) quantifier followed by at most alternation of quantifiers.